Nonlinear Elastic Model Research Articles

Deformation localisation in stretched liquid crystal elastomers

We model within the framework of finite elasticity two inherent instabilities observed in liquid crystal elastomers under uniaxial tension. First is necking, which occurs when a material sample suddenly elongates more in a small region where it appears narrower than the rest of the sample. Second is shear striping, which forms when the in-plane director rotates gradually to realign and become parallel with the applied force. These phenomena are due to the liquid crystal molecules rotating freely under mechanical loads. To capture necking, we assume that the uniaxial order parameter increases with tensile stretch, as reported experimentally during polydomain-monodomain transition. To account for shear striping, we maintain the uniaxial order parameter fixed, as suggested by experiments. Our finite element simulations capture well these phenomena. As necking in liquid crystal elastomers has not been satisfactorily modelled before, our theoretical and numerical findings related to this effect can be of wide interest. Shear striping has been well studied, yet our computed examples also show how optimal stripe width increases with the nematic penetration depth measuring the competition between the Frank elasticity of liquid crystals and polymer elasticity. Although known theoretically, this result has not been confirmed numerically by previous nonlinear elastic models.

Novel results from quadratically nonlinear elastic wave models using Murnaghan’s potential

AbstractIn this article, we study one, two and three-dimensional nonlinear elastic wave equations using quadratically nonlinear Murnaghan potential. We employ two effective methods for obtaining approximate series solutions the Adomian decomposition and the variational iteration method. These methods have the advantage of not requiring any physical parametric assumptions in the problem. Finally, these methods can generate expansion solutions for linear and nonlinear differential equations without perturbation, linearization, or discretization. The results obtained using the adopted methods along various initial and boundary conditions are in excellent agreement with the numerical results on MATLAB, which show the reliability of our methods to these problems. We came to the conclusion that our methods are accurate and simple to use.

Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model

For nonlinear Cosserat elasticity, we consider multiscale methods in this paper. In particular, we explore the generalized multiscale finite element method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting property (ensuring bounded linearized strains even under high stresses). Such strain-limiting Cosserat model can find potential applications in solids and biological fibers. However, Cosserat media with naturally rotational degrees of freedom, nonlinear constitutive relations, high contrast, and heterogeneities may produce challenging multiscale characteristics in the solution, and upscaling by multiscale methods is necessary. Therefore, we utilize the offline and residual-based online (adaptive or uniform) GMsFEM in this context while handling the nonlinearity by Picard iteration. Through various two-dimensional experiments (for perforated, composite, and stochastically heterogeneous media with small and big strain-limiting parameters), our numerical results show the approaches' convergence, efficiency, and robustness. In addition, these results demonstrate that such approaches provide good accuracy, the online GMsFEM gives more accurate solutions than the offline one, and the online adaptive strategy has similar accuracy to the uniform one but with fewer degrees of freedom.

Analytical model of friction at low shear rates for soft materials in 3D printing.

The biological properties of silicone elastomers such as polydimethylsiloxane (PDMS) have widespread use in biomedicine for soft tissue implants, contact lenses, soft robots, and many other small medical devices, due to its exceptional biocompatibility. Additive manufacturing of soft materials still has significant challenges even with major advancements that have occurred in development of these technologies for customized medical devices and tissue engineering. The aim of this study was to develop a mathematical model of tangential stress in relation to shear stress, shear rate, 3D printing pressure and velocity, for non-Newtonian gels and fluids that are used as materials for 3D printing. This study used FENE (finitely extensible nonlinear elastic model) model, for non-Newtonian gels and fluids to define the dependences between tangential stress, velocity, and pressure, considering viscosity, shear stress and shear rates as governing factors in soft materials friction and adhesion. Experimental samples were fabricated as showcases, by SLA and FDM 3D printing technologies: elastic polymer samples with properties resembling elastic properties of PDMS and thermoplastic polyurethane (TPU) samples. Experimental 3D printing parameters were used in the developed analytical solution to analyse the relationships between governing influential factors (tangential stress, printing pressure, printing speed, shear rate and friction coefficient). Maple software was used for numerical modelling. Analytical model applied on a printed elastic polymer, at low shear rates, exhibited numerical values of tangential stress of 0.208-0.216 N m - 2 at printing velocities of 0.9 to 1.2 mm s - 1, while the coefficient of friction was as low as 0.09-0.16. These values were in accordance with experimental data in literature. Printing pressure did not significantly influence tangential stress, whereas it was slightly influenced by shear rate changes. Friction coefficient linearly increased with tangential stress. Simple analytical model of friction for elastic polymer in SLA 3D printing showed good correspondence with experimental literature data for low shear rates, thus indicating possibility to use it for prediction of printing parameters towards desired dimensional accuracy of printed objects. Further development of this analytical model should enable other shear rate regimes, as well as additional soft materials and printing parameters.

An improved viscoelastic k- ε -v2-f turbulence model for intermediate and high drag reduction regimes

In this work, an improved anisotropic k-ε-v2-f model based on the finite extensible nonlinear elastic model with the Peterlin approximation for viscoelastic channel flows is proposed. This model is tested using direct numerical simulation (DNS) data for friction Reynolds numbers (Reτ) in the range of 120–1000, friction Wiesenberg numbers (Wiτ) in the range of 25–116, viscosity ratios (β) in the range of 0.6–0.9, and maximum polymer extensibility values (L2) in the range of 900–14 400. The flow characteristics of viscoelastic fluids with various parameters obtained from the new model agree well with existing DNS results. By adding closures for the flow, shear, and transverse components, the incomplete prediction of nonlinear terms from interactions between the fluctuating components of the conformation and velocity gradient tensors is improved. Compared with DNS results, these closures can fully obtain each component and significantly improve the accuracy of the flow direction component in the intermediate and high drag reduction regimes. Furthermore, the model in this paper retains the advantages of an anisotropic model, does not require a damping function, is simple to construct, and is easily extended to a variety of bounded flows.

Calculation of 3D Shape of a Hyperelastic Body for Nonlinear Elasticity Models Using the Newton Method

This article discusses methods for calculating the deformations of objects with hyperelastic materials within the framework of the nonlinear elasticity theory. This topic is relevant due to the use of new technological materials in industry, and as a result, the emerging task of preliminary numerical calculations for operational reliability, research and modeling of deformation behavior. The first part provides a brief overview of the main provisions and formulas of the nonlinear elasticity theory. Then the spatial discretization of 3-dimensional objects and the calculation of the deformation gradient using the finite element method are considered. The article provides an algorithm for solving this problem, such as minimizing the functional of stored energy, and also considers the class of permissible deformations. Afterwards, a detailed description is given, along with pseudocode, of the method of implementing and calculating the minimization by Newton’s method. The last part demonstrates an example of the calculations performed, based on the developed software that allows for numerical experiments and computer modeling of deformations of hyperelastic bodies, and one of the capabilities of which is to perform these calculations using Newton’s method.

Study on fragmentation characteristics of carbon nanotube-enhanced concrete and a comparative evaluation of the ZWT and ottosen constitutive model

In order to study the difference between the crushing characteristics of carbon nanotubes (CNTs) concrete and ordinary concrete, this paper conducts a test between ordinary concrete and CNTs concrete with a volume content of 0.3 % at different impact speeds through the Hopkinson test, using fragmentation and particle size distribution as evaluation indexes. Crushed fragments are analyzed from the perspective of concrete pores. Finally, the nonlinear viscoelastic constitutive model and the nonlinear elastic constitutive model of CNTs concrete are then developed and compared. Based on the results, the average pores of the CNTs concrete decreased from 42.0 nm to 38.0 nm, but the pore area increased from 5.743 m2/g to 6.305 m2/g, indicating that CNTs can fill the internal pores. Meanwhile, the average particle size of broken CNTs concrete is significantly larger than that of ordinary concrete. However, the relationship between the crushing particle size and impact velocity is approximately y=A-Bln(x+C) for both concretes. According to the study of concrete constitutive models, compared to the Loand damage model, the combination of Loand model and Mazars model is more accurate in describing concrete damage characteristics, and the average correlation coefficient increases from 0.84 to 0.92. Meanwhile, compared to the Zhu-Wang-Tang (ZWT) model, Ottosen model can better capture the dynamic mechanical properties of CNT concrete, with a correlation coefficient over 0.98.

Displacement and pressure of surrounding rock during shield tunnelling and supporting in low water content loess

Using a self-developed micro shield machine, a laboratory-scale experimental scheme for shield tunnelling and supporting was designed, and the soil displacement law and earth pressure on the segments during tunnel construction were studied. Based on a nonlinear elastic constitutive model describing the characteristics of loess, the UMAT subroutine was redeveloped and verified by conventional triaxial tests. It was then applied to the numerical simulation of shield tunnelling and supporting. The following are the findings: 1) The disturbance modes of surrounding rock directly above the shield tunnel under different boundaries are similar. The main disturbances occur approximately 4 m above the shield tunnel, extending to a distance of one segment outer diameter on either side of the tunnel's central axis. 2) The pressures on the top and bottom of the segments can be divided into rapid increase, slow increase, and stable fluctuation. These stages correspond to the processes of the segments getting out of the shield shell and the soil collapsing. 3) The largest surrounding rock pressure is at the vault, followed by that at the arch bottom, and the smallest is at both sides of the arch waist, which is consistent with the law obtained based on laboratory-scale model tests. The surrounding rock pressure values of the vault in the numerical simulation are consistent with the laboratory-scale model test results. These verify the reliability of the laboratory-scale model test and the UMAT subroutine based on the nonlinear elastic constitutive model of loess in simulating the shield tunnel construction in loess strata.

Finite element modelling of pipe piles driven in low-to-medium density chalk under monotonic axial loading

Percussive driving in low-to-medium density chalk creates a thin annulus of fully de-structured ‘putty’ chalk around pile shafts and an outer annular zone where fracturing is more intense than in the natural chalk. This damage and the related generation, and subsequent equalisation, of excess pore pressures impacts the piles’ time-dependent axial loading behaviour, as do other ageing processes. This paper presents finite element analyses of open steel tubular piles driven for the recent ALPACA and ALPACA Plus research projects under monotonic axial loading to failure after extended ageing periods. A nonlinear elastic stiffness model with a nonlocal deviatoric strain-based Mohr-Coulomb failure criterion was employed, with different sets of properties to represent the de-structured, fractured chalk and intact chalk. It is shown that the piles’ axial responses are mainly controlled by the puttified chalk annuli. A simplified but efficient means is adopted to impose pre-loading chalk effective stress conditions that explicitly capture the effects of installation damage and subsequent ageing. The potential for strain-softening in the brittle chalk is examined and the effective stress paths developed in representative chalk elements throughout the loading are considered in conjunction with the mobilisation of accumulated deviatoric strains. The simulations indicate 264% higher shaft capacity in compression than in tension, which is mainly attributed to the internal chalk plug.

Effect of steel fibers and the fiber-concrete adhesion stress on the nonlinear shear behavior of beams

In this article, a theoretical model in nonlinear elasticity is presented to analyze the influence of the percentage of steel fibers and the effect of the fiber-concrete bond stress on the shear force at the rupture of beams subjected to the combined effect of bending moment, normal force, and shear force. For a given beam section, it is defined by a succession of layers of concrete and longitudinal steel elements. Each layer is defined by its height hi, width bi, and position relative to one end of the section YGi. Each longitudinal steel element is also defined by its cross-sectional area and position relative to one end of the section. The steel fibers are defined by volume percentages of 0.5%, 1%, 1.5%, and 2%, taking into account the mechanical nonlinearity of the materials. This model is based on the multilayer analysis of sections and an iterative solution procedure for each layer and each section, considering a given longitudinal deformation state and shear stress. The global equilibrium of the sections is analyzed under the assumption of flat longitudinal deformations but with, in principle, an interdependence of longitudinal normal stresses and shear stresses. In this study, using the principle of virtual work, equilibrium equations for deformations and stresses, as well as partial compatibility equations between concrete deformations and mean deformations, are derived. Comparative examples between ordinary reinforced concrete beams with variable shapes and reinforcement details, and those reinforced with steel fibers, are presented to demonstrate the accuracy of the proposed model for simulating the nonlinear shear response of beams.

Multiscale simulations of viscoelastic fluids in complex geometries using a finitely extensible nonlinear elastic transient network model

This paper presents a novel implementation of a numerical scheme for predicting complex flows of viscoelastic fluids using a finitely extensible nonlinear elastic (FENE) transient network model. This model extends the FENE model by incorporating chain interactions and accounting for the way in which the maximum chain length, drag, and relaxation time are influenced by entanglement and disentanglement processes. Three different initial networks are considered (disentanglement, entanglement, and aleatory), and the influence of variables such as the kinetic rate constants, elasticity, and chain length on the microstate concentration, stresses, and drag force is investigated. It is shown that although the concentrations of the microstates are independent of the Weissenberg number and the maximum extension length, the stresses and hence the drag are influenced by them.

Anisotropic Damage Mechanism of Coal Seam Water Injection with Multiphase Coupling.

After coal seam water injection, coal mechanical properties will change with brittleness weakening and plasticity enhancement. Aiming at the problem of coal damage caused by the coal seam water injection process, based on nonlinear pore elasticity theory and continuum damage theory, a nonlinear pore elastic damage model considering anisotropic characteristics is proposed to calculate and analyze the gas-liquid-solid multiphase coupling effect with the fully coupled finite element method during the coal seam water injection process. The research results indicate that the wetting radius of calculated results by the model agrees well with the in situ test results, and the relative errors are less than 10%. Water saturation and induced damage of the coal body in the parallel bedding direction are greater than that in the vertical bedding direction during the coal seam water injection process, which exhibits significant anisotropic characteristics. With the increasing water injection time, the induced damage of the coal body also increases near the water injection hole. Considering the inherent permeability arising with damage, it has a significant impact on both water saturation and induced damage, which also indicates that there is a strong interaction between water saturation and induced damage. The theoretical model reveals the coal damage mechanism of gas-liquid-solid multiphase coupling after coal seam water injection and provides a theoretical prediction of coal containing water characteristics in engineering practice.

Evaluating Different Track Sub-Ballast Solutions Considering Traffic Loads and Sustainability

The railway industry is seeking high-performance and sustainable solutions for sub-ballast materials, particularly in light of increasing cargo transport demands and climate events. The meticulous design and construction of track bed geomaterials play a crucial role in ensuring an extended track service life. The global push for sustainability has prompted the evaluation of recycling ballast waste within the railway sector, aiming to mitigate environmental contamination, reduce the consumption of natural resources, and lower costs. This study explores materials for application and compaction using a formation rehabilitation machine equipped with an integrated ballast recycling system designed for heavy haul railways. Two recycled ballast-stabilised soil materials underwent investigation, meeting the necessary grain size distribution for the proper compaction and structural conditions. One utilised a low-bearing-capacity silty sand soil stabilised with recycled ballast fouled waste (RFBW) with iron ore at a 3:7 weight ratio, while the second was stabilised with 3% cement. Laboratory tests were conducted to assess their physical, chemical, and mechanical properties, and a non-linear elastic finite element numerical model was developed to evaluate the potential of these alternative solutions for railway sub-ballast. The findings indicate the significant potential of using soils stabilised with recycled fouled ballast as sub-ballast for heavy haul tracks, underscoring the advantages of adopting sustainable sub-ballast solutions through the reuse of crushed deteriorated ballast material.

Nonlinear consolidation finite element analysis of a layered soft soil foundation under multistage loading based on the continuous drainage boundary

To comprehensively consider the impacts of stratification, residual pore water pressure, soil nonlinearity, and boundary permeability on consolidation settlement of soft soil foundations for accurate prediction, a continuous drainage boundary condition is proposed in this study that reflects the residual pore pressure under multistage loading, and a nonlinear elastic constitutive model based on the double logarithmic model is adopted to account for the nonlinear consolidation behaviour of soils. A UMAT subroutine is developed based on the proposed boundary condition and nonlinear elastic constitutive model. Subsequently, the developed subroutine is compared with the built-in linear elastic soil constitutive model in ABAQUS and engineering examples. The application of continuous drainage boundaries in stratified foundations is analysed, as well as the influence of factors such as the loading rate and soil nonlinearity on consolidation settlement. The results indicate that, compared to the built-in model, the subroutine developed in this study can be employed to more accurately calculate the nonlinear consolidation of multilayered foundations under multistage loading. By adjusting the loading rate parameter αk, consolidation under different loading conditions can be predicted. Additionally, the proposed boundary condition simplifies the calculations for soft soil foundations with sand layers, providing a novel computational approach for the design of construction loading schemes and long-term settlement predictions in soft soil foundations.

Laboratory Test and Constitutive Model for Quantifying the Anisotropic Swelling Behavior of Expansive Soils

Expansive soils exhibit directionally dependent swelling that traditional isotropic models fail to capture. This study investigates the anisotropic swelling characteristics of expansive soil with a medium swelling potential through the use of modified oedometric testing. Vertical swelling strains can reach up to 1.71 times that of the horizontal movements, confirming intrinsic anisotropy. A nonlinear elastic constitutive model incorporates vertical and horizontal elastic moduli with respect to matric suction to characterize anisotropy. Three elastic parameters were determined through the experiments, and predictive equations were developed to estimate the unsaturated moduli. The constitutive model and predictive techniques provide practical tools to better assess expansive soil pressures considering anisotropy, offering guidelines for utilization and design. The outcomes advance understanding of these soils’ directionally dependent behavior and stress–strain–suction response.

Non-linear behaviour of soil–pile interaction phenomena and its effect on the seismic response of OWT pile foundations. Validity range of a linear approach through non-degraded soil properties

The effect of non-linear and inelastic behaviour of the soil–pile interaction on the seismic response of offshore wind turbine pile foundations embedded in sandy soils is analysed. For this purpose, the responses obtained assuming three different Beam on Dynamic Winkler Foundation (BDWF) models are compared: a Plastic Non-Linear Model (PNLM), an Elastic Non-Linear Model (ENLM) and a simple elastic linear model with non-degraded properties of soil (NDLM). The influence of non-linearity and plasticity assumption is studied by evaluating the effects of the kinematic and inertial interaction within soil–pile interaction. Two soil stiffness levels are analysed: very loose and medium dense sand. The seismic response under ten earthquakes is computed in terms of mean envelopes of internal forces along the pile length. The non-linearity and inelastic influence of soil–pile interaction is quantified by means of relative differences with respect to the linear elastic model. Results show that the non-linear and inelastic models acquire relevance when the contribution of the inertial interaction dominates, leading to lower maximum responses than the linear elastic model. If the inertial interaction is not significantly activated, similar results between the three models are obtained, being the linear elastic model enough to reproduce the soil–pile dynamic interaction.

Inertial enhancement of the polymer diffusive instability

Beneitez et al. (Phys. Rev. Fluids, vol. 8, 2023, L101901) have recently discovered a new linear ‘polymer diffusive instability’ (PDI) in inertialess rectilinear viscoelastic shear flow using the finitely extensible nonlinear elastic constitutive model of Peterlin (FENE-P) when polymer stress diffusion is present. Here, we examine the impact of inertia on the PDI for both plane Couette and plane Poiseuille flows under varying Weissenberg number ${W}$ , polymer stress diffusivity $\varepsilon$ , solvent-to-total viscosity ratio $\beta$ and Reynolds number ${Re}$ , considering the FENE-P and simpler Oldroyd-B constitutive relations. Both the prevalence of the instability in parameter space and the associated growth rates are found to significantly increase with ${Re}$ . For instance, as $Re$ increases with $\beta$ fixed, the instability emerges at progressively lower values of $W$ and $\varepsilon$ than in the inertialess limit, and the associated growth rates increase linearly with $Re$ when all other parameters are fixed. For finite $Re$ , it is also demonstrated that the Schmidt number $Sc=1/(\varepsilon Re)$ collapses curves of neutral stability obtained across various $Re$ and $\varepsilon$ . The observed strengthening of PDI with inertia and the fact that stress diffusion is always present in time-stepping algorithms, either implicitly as part of the scheme or explicitly as a stabilizer, implies that the instability is likely operative in computational work using the popular Oldroyd-B and FENE-P constitutive models. The fundamental question now is whether PDI is physical and observable in experiments, or is instead an artifact of the constitutive models that must be suppressed.

Modelling rapid non-destructive test using light weight deflectometer on granular soils across different degrees of saturation

This study introduces an advanced finite element model for the light weight deflectometer (LWD), which integrates contact mechanics with fully coupled models. By simulating LWD tests on granular soils at various saturation levels, the model accurately reflects the dependence of the LWD modulus on dry density, water content, and effective stress. This model addresses and overcomes the limitations of previous finite element models for this specific problem. Simultaneously, this research presents the first experimentally validated fully coupled contact impact model. Furthermore, the research provides a comparative assessment of elastoplastic and nonlinear elastic models and contrasts an enriched node-to-segment method (developed in this study) with the more precise mortar technique for contact mechanics. These comparisons reveal unique advantages and challenges for each method. Moreover, the study underscores the importance of careful application of the LWD modulus, emphasising the need for sophisticated tools to interpret soil behaviour accurately.

Generalized Multiscale Finite Element Treatment of a Heterogeneous Nonlinear Strain-limiting Elastic Model

Generalized Multiscale Finite Element Treatment of a Heterogeneous Nonlinear Strain-limiting Elastic Model

Application and reduction of a nonlinear hyperelastic wall model capturing ex vivo relationships between fluid pressure, area, and wall thickness in normal and hypertensive murine left pulmonary arteries.

Pulmonary hypertension is a cardiovascular disorder manifested by elevated mean arterial blood pressure (>20 mmHg) together with vessel wall stiffening and thickening due to alterations in collagen, elastin, and smooth muscle cells. Hypoxia-induced (type 3) pulmonary hypertension can be studied in animals exposed to a low oxygen environment for prolonged time periods leading to biomechanical alterations in vessel wall structure. This study introduces a novel approach to formulating a reduced order nonlinear elastic structural wall model for a large pulmonary artery. The model relating blood pressure and area is calibrated using ex vivo measurements of vessel diameter and wall thickness changes, under controlled pressure conditions, in left pulmonary arteries isolated from control and hypertensive mice. A two-layer, hyperelastic, and anisotropic model incorporating residual stresses is formulated using the Holzapfel-Gasser-Ogden model. Complex relations predicting vessel area and wall thickness with increasing blood pressure are derived and calibrated using the data. Sensitivity analysis, parameter estimation, subset selection, and physical plausibility arguments are used to systematically reduce the 16-parameter model to one in which a much smaller subset of identifiable parameters is estimated via solution of an inverse problem. Our final reduced one layer model includes a single set of three elastic moduli. Estimated ranges of these parameters demonstrate that nonlinear stiffening is dominated by elastin in the control animals and by collagen in the hypertensive animals. The pressure-area relation developed in this novel manner has potential impact on one-dimensional fluids network models of vessel wall remodeling in the presence of cardiovascular disease.